Czasopismo
Tytuł artykułu
Warianty tytułu
Języki publikacji
Abstrakty
We announce some results on compactifying moduli spaces of rank 2 vector bundles on surfaces by spaces of vector bundles on trees of surfaces. This is thought as an algebraic counterpart of the so-called bubbling of vector bundles and connections in differential geometry. The new moduli spaces are algebraic spaces arising as quotients by group actions according to a result of Kollár. As an example, the compactification of the space of stable rank 2 vector bundles with Chern classes c 1 = 0, c 1 = 2 on the projective plane is studied in more detail. Proofs are only indicated and will appear in separate papers.
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
1331-1355
Opis fizyczny
Daty
wydano
2012-08-01
online
2012-05-31
Twórcy
autor
- Université Lille 1, markushe@math.univ-lille1.fr
autor
- Yaroslavl State Pedagogical University, astikhomirov@mail.ru
autor
- Universität Kaiserslautern, trm@mathematik.uni-kl.de
Bibliografia
- [1] Barth W., Moduli of vector bundles on the projective plane, Invent. Math., 1977, 42 63–91 http://dx.doi.org/10.1007/BF01389784
- [2] Buchdahl N.P., Sequences of stable bundles over compact complex surfaces, J. Geom. Anal., 1999, 9(3), 391–428 http://dx.doi.org/10.1007/BF02921982
- [3] Buchdahl N.P., Blowups and gauge fields, Pacific J. Math., 2000, 196(1), 69–111 http://dx.doi.org/10.2140/pjm.2000.196.69
- [4] Donaldson S.K., Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. London Math. Soc., 1985, 50(1), 1–26 http://dx.doi.org/10.1112/plms/s3-50.1.1
- [5] Donaldson S.K., Compactification and completion of Yang-Mills moduli spaces, In: Differential Geometry, Peñíscola, 1988, Lecture Notes in Math., 1410, Springer, Berlin, 1989, 145–160
- [6] Feehan P.M.N., Geometry of the ends of the moduli space of anti-self-dual connections, J. Differential Geom., 1995, 42(3), 465–553
- [7] Fulton W., MacPherson R., A compactification of configuration spaces, Ann. of Math., 1994, 139(1), 183–225 http://dx.doi.org/10.2307/2946631
- [8] Gieseker D., On the moduli of vector bundles on an algebraic surface, Ann. of Math., 1977, 106(1), 45–60 http://dx.doi.org/10.2307/1971157
- [9] Gieseker D., A construction of stable bundles on an algebraic surface, J. Differential Geom., 1988, 27(1), 137–154
- [10] Huybrechts D., Lehn M., The Geometry of Moduli Spaces of Sheaves, 2nd ed., Cambridge Math. Lib., Cambridge University Press, Cambridge, 2010 http://dx.doi.org/10.1017/CBO9780511711985
- [11] Kirwan F.C., Partial desingularisations of quotients of nonsingular varieties and their Betti numbers, Ann. of Math., 1985, 122(1), 41–85 http://dx.doi.org/10.2307/1971369
- [12] Kollár J., Projectivity of complete moduli, J. Differential Geom., 1990, 32(1), 235–268
- [13] Kollár J., Quotient spaces modulo algebraic groups, Ann. of Math., 1997, 145(1), 33–79 http://dx.doi.org/10.2307/2951823
- [14] Li J., Algebraic geometric interpretation of Donaldson’s polynomial invariants, J. Differential Geom., 1993, 37(2), 417–466
- [15] Lübke M., Teleman A., The Kobayashi-Hitchin Correspondence, World Scientific, River Edge, 1995 http://dx.doi.org/10.1142/2660
- [16] Maruyama M., Singularities of the curve of jumping lines of a vector bundle of rank 2 on ℙ2, In: Algebraic Geometry, Tokyo, Kyoto, October 5–14, 1982, Lecture Notes in Math., 1016, Springer, Berlin-New York, 1983, 370–411
- [17] Maruyama M., Trautmann G., On compactifications of the moduli space of instantons, Internat. J. Math., 1990, 1(4), 431–477 http://dx.doi.org/10.1142/S0129167X90000228
- [18] Maruyama M., Trautmann G., Limits of instantons, Internat. J. Math., 1992, 3(2), 213–276 http://dx.doi.org/10.1142/S0129167X92000072
- [19] Nagaraj D.S., Seshadri C.S., Degenerations of the moduli spaces of vector bundles on curves. I, Proc. Indian Acad. Sci. Math. Sci., 1997, 107(2), 101–137
- [20] Nagaraj D.S., Seshadri C.S., Degenerations of the moduli spaces of vector bundles on curves. II, Proc. Indian Acad. Sci. Math. Sci., 1999, 109(2), 165–201 http://dx.doi.org/10.1007/BF02841533
- [21] Okonek C., Schneider M., Spindler H., Vector Bundles on Complex Projective Spaces, Progr. Math., 3, Birkhäuser, Boston, 1980
- [22] Simpson C.T., Moduli of representations of the fundamental group of a smooth projective variety I, Inst. Hautes Études Sci. Publ. Math., 1994, 79, 47–129 http://dx.doi.org/10.1007/BF02698887
- [23] Taubes C.H., A framework for Morse theory for the Yang-Mills functional, Invent. Math., 1988, 94(2), 327–402 http://dx.doi.org/10.1007/BF01394329
- [24] Timofeeva N.V., Compactification of the moduli variety of stable 2-vector bundles on a surface in the Hilbert scheme, Math. Notes, 2007, 82(5–6), 667–690
- [25] Timofeeva N.V., On a new compactification of the moduli of vector bundles on a surface, Sb. Math., 2008, 199(7–8), 1051–1070 http://dx.doi.org/10.1070/SM2008v199n07ABEH003953
- [26] Timofeeva N.V., On a new compactification of the moduli of vector bundles on a surface. II, Sb. Math., 2009, 200(3–4), 405–427 http://dx.doi.org/10.1070/SM2009v200n03ABEH004002
- [27] Timofeeva N.V., On a new compactification of the moduli of vector bundles on a surface. III: A functorial approach, Sb. Math., 2011, 202(3–4), 413–465 http://dx.doi.org/10.1070/SM2011v202n03ABEH004151
- [28] Trautmann G., Moduli spaces in algebraic geometry (manuscript)
- [29] Uhlenbeck K.K., Removable singularities in Yang-Mills fields, Comm. Math. Phys., 1982, 83(1), 11–29 http://dx.doi.org/10.1007/BF01947068
- [30] Viehweg E., Quasi-Projective Moduli for Polarized Manifolds, Ergeb. Math. Grenzgeb., 30, Springer, Berlin, 1995 http://dx.doi.org/10.1007/978-3-642-79745-3
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-012-0072-0